Skip to content
Carmelics
Topics
Thinkers
Changes
Contributors
Loading account…
Statements
321,452
Perspectives
108,905
Topics
42
Home
/
Original
/
inverse
See Original
Inverse View
It is not the case that If there exist sentences φ such that Γ ⊨ φ yet Γ ⊬ φ within sufficiently expressive systems, the universal claim Γ ⊢ φ is false as a general principle.
?
Set your confidence on the premises below to see your aggregate.
Reasons For
1 perspective
Reason for
?
1.
The distinction between ⊨ and ⊢ is system-relative; a claim true in one proof system may be derivable in a stronger one, so no universal failure exists.
?
How convincing is this?
Think about whether this reason is strong or weak
2.
Gödel's results apply to *specific* unprovable sentences, not all entailed sentences—most logically valid inferences remain provable in standard systems.
?
How convincing is this?
Think about whether this reason is strong or weak
3.
Rejecting 'Γ ⊢ φ universally' based on edge cases conflates 'not always provable' with 'the principle is false,' when it may just need qualification.
?
How convincing is this?
Think about whether this reason is strong or weak
Reasons Against
1 perspective
Reason against
?
1.
Semantic entailment (⊨) captures what must be true across all models; syntactic derivability (⊢) is bounded by proof systems and their axioms.
?
How convincing is this?
Think about whether this reason is strong or weak
2.
Gödel's incompleteness shows sufficiently expressive systems have truths unprovable within them, demonstrating the gap between ⊨ and ⊢.
?
How convincing is this?
Think about whether this reason is strong or weak
3.
If Γ ⊨ φ but Γ ⊬ φ exists as a concrete case, claiming Γ ⊢ φ universally contradicts that evidence and overstates deductive power.
?
How convincing is this?
Think about whether this reason is strong or weak
Next step
Based on where you are in your exploration
Strongest counterpoint
Explore the most compelling reason on the other side.